For \(|F_\alpha |^2\) , setting \({\cal F}_\alpha = {\cal F}_\beta = {\cal G}_\gamma = {\cal G}_\delta \)  in (REF ) and using \(\langle {\cal F}^2_\alpha \rangle = 0\)  owing to the complex circularity (\(\langle {\cal E}^2_0 \rangle = 0\) ), this yields
\(\langle |{\cal F}_\alpha |^4 \rangle =2 \langle |{\cal F}_\alpha |^2 \rangle ^2+|\langle {\cal F}^2_\alpha \rangle |^2=2 \langle |{\cal F}_\alpha |^2 \rangle ^2\) ,
leading to the familiar result
\(\sigma _{|F_\alpha |^2} =\langle |F_\alpha |^2 \rangle \)  for \(\alpha =x,y,z\) 
[1]}.
For the energies, this results in explicit functions of \(kz\)  as
\(\sigma _{U_{\alpha }}(kz) &= \left( 1 \mp j_0 \pm \frac{j_2}{2} \right) \sigma _{U_{\alpha ,\infty }},~~ \alpha =x,y\\\sigma _{U_{z}}(kz) &= \left( 1 \pm j_0 \pm j_2 \right) \sigma _{U_{\alpha ,\infty }}\) 
